Berkovich analytic spaces are a version of analytic spaces over nonarchimedean fields. Unlike the rigid analytic spaces (see rigid analytic geometry) of Tate, which are locally defined via maximal spectra of Tate algebras glued via the Grothendieck G-topology, the Berkovich analytic spaces are actual topological space equipped with a cover by affinoid domains via the analytic spectrum construction, due to Vladimir Berkovich. This spectrum can be viewed as consisting of the data of prime ideal plus the extension of the norm to the residue field; thus the Berkovich spectrum has far more points (though fewer than, say, Huber's adic spaces which may also contain valuations of higher order).
For more background see analytic geometry.
Definition of Berkovich analytic spaces
Let be a non-archimedean field.
Given and positive elements , consider the sub-power series algebra over of those series which converge inside the radii , i.e. the algebra defined by
This is a commutative Banach algebra over with norm .
A -affinoid algebra is a commutative Banach -algebra for which there exists and as above and an epimorphism
such that the norm on is the quotient norm.
If one can choose here for all then is called strictly -affinoid.
The category of -affinoid spaces is the opposite category of the category of -affinoid algebras and bounded homomorphisms between them.
Via the analytic spectrum there is a topological space associated with any -affinoid space. Often this underlying topological space is referred to as the analytic space.
An affinoid domain in an affinoid space is a closed subset such that there is a homomorphism of -affinoid spaces
for some , whose image is , and such that every other morphism of -affinoid spaces into whose image is contained in uniquely factors through this morphism.
(Berkovich 09, def. 3.1.4)
Under some mild conditions, the algebraic and the analytic étale cohomology of Berkovich spaces agree. (Berkovich 95)
The underlying topological space given by the Berkovich analytic spectrum has as singular cohomology the weight 0-cohomology of (Berkovich 09).
See also MO discussion here.
A complex analytic manifold and a smooth complex analytic space is locally isomorphic to a polydisk and hence is trivially a locally contractible space. But over a non-archimedean field analytic spaces no longer need to be locally isomorphic to polydisks (but -adic polydisks are still contractible (Berkovich 90)). The following result establishes, under mild conditions, that general analytic spaces are nevertheless locally contractible.
Assume that the valuation on the ground field is nontrivial.
A -analytic space is called locally embeddable in a smooth space if each point of has an open neighbourhood isomorphic to a strictly -analytic domain in smooth -analytic space.
Every -analytic space which is locally embeddable in a smooth space, def. 4, is a locally contractible space.
More precisely, every point of a locally smooth -analytic space has an open neighbourhood which is contractible, and which is a union of analytic domains.
The local contractibility is Berkovich (1999), theorem 9.1. The refined statment in terms of inductive systems of analytic domains is in Berkovich (2004).
- The proof of the local Langlands conjecture for by Harris–Taylor uses étale cohomology on non-archimedean analytic spaces (in the sense of Berkovich) to construct the required Galois representations over local fields.
Introductions and reviews
A nice survey is in
- Bernard Le Stum, One century of -adic geometry – From Hensel to Berkovich and beyond, talk notes, June 2012 (pdf)
A good introduction to the general idea is at the beginning of
Basic notions are listed in
- M. Temkin, Non-archimedean analytic spaces (pdf slides)
A review of basic definitions and facts about affinoid and rigid -analytic spaces can be found in
- Gaëtan Chenevier, lecture 5 (pdf)
See also the references at rigid analytic geometry.
A review of definitions and results on -analytic spaces is in
- Vladimir Berkovich, -Adic analytic spaces, in Proceedings of the International Congress of Mathematicians, Berlin, August 1998, Doc. Math. J. DMV, Extra Volume ICM II (1998), 141-151 (pdf)
A more detailed set of lecture notes along these lines is
- Vladimir Berkovich, Non-archimedean analytic spaces, lectures at the Advanced School on -adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)
Introductory exposition of the Berkovich analytic spectrum is
Jérôme Poineau, Global analytic geometry, pages 20-23 in EMS newsletter September 2007 (pdf)
Frédéric Paugam, section 2.1.4 of_Global analytic geometry and the functional equation_ (2010) (pdf)
A exposition of examples of Berkovich spectra is in
Vladimir Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, (1990) 169 pp.
Vladimir Berkovich, Étale cohomology for non-Archimedean analytic spaces, Publ. Math. IHES 78 (1993), 5-161.
Discussion of Berkovich09cohomology of Berkovich analytic spaces includes
Vladimir Berkovich, On the comparison theorem for étale cohomology of non-Archimedean analytic spaces. Israel Journal of Mathematics 92.1-3 (1995): 45-59.
Vladimir Berkovich, A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures, Algebra, Arithmetic, and Geometry. Birkhäuser Boston, 2009. 49-67.
Discussion of local contractibility of smooth -analytic spaces is in
Vladimir Berkovich, Smooth -adic analytic spaces are locally contractible, Invent. Math. 137 1-84 (1999) (pdf)
Vladimir Berkovich, Smooth p-adic analytic spaces are locally contractible. II, in Geometric Aspects of Dwork Theory, Walter de Gruyter & Co., Berlin, (2004), 293-370. (pdf)
and more generally in
Relation to other topics
On the relation to buildings:
- Annette Werner, Buildings and Berkovich Spaces (pdf)
Relation to integration theory
- Vladimir Berkovich, Integration of 1-forms on -adic analytic spaces, Princeton University Press,
Aspects of the homotopy theory/étale homotopy of analytic spaces are discussed in
- Aise Johan de Jong, Étale fundamental groups of non-archimedean analytic spaces, Mathematica, 97 no. 1-2 (1995), p. 89-118 (numdam)
Relation to formal schemes:
Discussion of Berkovich analytic geometry as algebraic geometry in the general sense of Bertrand Toën and Gabriele Vezzosi is in